Hasse-Weil inequality and primality tests in the context of curves of genus 2

University of Groningen

Hasse-Weil inequality and primality tests in the context of curves of genus 2

Ruíz Duarte, Eduardo

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Publication date: 2018

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Ruíz Duarte, E. (2018). Hasse-Weil inequality and primality tests in the context of curves of genus 2. Rijksuniversiteit Groningen.

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Stellingen

“Hasse-Weil Inequality and Primality Tests in the

context of Curves of Genus 2”

Eduardo Ru´ız Duarte.

In Propositions 2, 3, 4, and 5 below, the following notations are used. Denote by H a hyperelliptic curve of genus g over Fq with q odd and J to its associated

Jacobian variety. We assume that the curve H is given by an equation Y2 ₌

f (X) with deg f (X) = 2g + 1. Let ∞ ∈ H be its unique point at infinity. Further, φ, [n] ∈ End(J ) denote the q-th Frobenius map and the multiplication by n map.

1. J.W.S. Cassels observed in a postscript of his review

(see https://mathscinet.ams.org/mathscinet-getitem?mr=81308) that some argument in Manin’s proof of the Hasse inequality for genus 1 is missing. A small change of Manin’s proof makes the statement that he did not prove, a triviality.

2. Let H have genus g and consider its generic point (x, y) ∈ H(Fq(x, y)) ∼=

H(Fq(H)). Consider its associated point g := [(x, y) − ∞] ∈ J . Further,

consider (φ + [−1])(g) = hu, vi with u, v ∈ Fq(H)[t] (Mumford

repre-sentation). Then d(v_dt2) _{∈ F}q(x) and x 7→ d(v2₎

dt is a rational function of

degree (2g − 1)q + #H(Fq) − 1.

3. Let H have genus 2 and let Θ be its image on J under the Abel-Jacobi map. Suppose that (φ + [n])(Θ) 6⊂ Θ. Take κ4∈ L(2Θ) ⊂ Fq(J )

(stan-dard basis element, as introduced in Proposition 2.4.3 of this thesis). With g as in the previous proposition, we have that κ4((φ + [n])(g)) =

µ1,n(x)

µ2,n(x) ∈ Fq(x) where the numerator and denominator are coprime

poly-nomials. Define δn := deg µ1,n(x). If each of δn−1, δn, and δn+1 are

defined, then 2δn+ 4 = δn−1+ δn+1. Moreover δn = 2n2+ (q + 1 −

#H(Fq))n + 2q. This implies the Hasse-Weil inequality for H.

4. Let H have genus 2 and consider the Abel-Jacobi map α : H → J . Sup-pose that there is an n ∈ Z such that ψn:= (φ + [n]) ◦ α ∈ Mor_Fq(H, J )

is constant. Then q is a perfect square and #H(Fq) = q + 1 + 4n =

q + 1 ± 4√q.

5. When mimicking Manin’s elementary proof of the Hasse inequality for genus 1 in the case of genus 2, we relied on the fact that the Theta divisor Θ in J is isomorphic to H. We controlled the behavior of (φ + [n])(H) ⊂ J which is generically a curve in J , hence an element in Div(J ). However, for g ≥ 3 this is not the case and then, we cannot work in Div(J ) anymore. This makes the existence of an elementary proof `a la Manin much less evident.

6. Consider the elliptic curve Et: y2= x3−(t2+1)x. Let m ≥ 1 be odd and

n > 1 an integer with 3, 56 | Am,n:= m2n− 1 with 4m < 2n. Let t ∈ Z

such that t2_{+ 1 is not a square modulo A}

m,n. Let (α_β,γ_δ) := m(−1, t) ∈

Et(Q) and define the sequence x0 := α_β, xi+1 := (x2

i+t 2₊₁₎2

4(x3

i−(t2+1)xi). The

integer Am,n is prime if and only if ximod Am,n is well defined for

0 ≤ i ≤ n − 1 and xn−1≡ 0 mod Am,n.

7. Let E30: 30y2= x3− x. Let p ≡ ±1 mod 10 be prime and consider the

integer Sp,n:= p216n+ 1 such that p < 2n. Let (5, 2) ∈ E30(Q(i)) and

consider the sequence defined by x0= 5 and xj+1:= i(1−x2_j)

2xj mod (p4

n₊

i). Then Sp,nis prime if and only if xjis well defined for all 0 ≤ j ≤ 4n−1

and x4n−1≡ 0 mod p4n+ i.

8. Let n > 1 be an odd integer and let λn := 4 · 5n− 1.

Consider the hyperelliptic curve H/Q(√5) given by y2_{= x}5_{+ h such that}

gcd(λn, h) = 1. The Jacobian J of H admits an endomorphism

√ 5 with square the multiplication by 5. Suppose F ∈ J (Q(√5)) is given and con-sider the sequence of divisors D0 := 4F, Di :=

√

5Di−1= hui(x), vi(x)i

with its coefficients reduced in Z[1+

√ 5 2 ]/(2 · 5 n+1 2 √ 5 − 1) ∼_{= Z/(λ}n).

If Dj is well defined and 6= h1, 0i for j ≤ 2n − 1 and D2n = h1, 0i then

λn is prime and F /∈ [

√

5]J (Fλn) for [

√

5] ∈ End_Fλn(J ).

9. It is sad and wonderful how humans create mathematics. We do this even knowing that our logic and lifetime is bounded by our nature of finite beings. The sad part is that we pursue the understanding of something too huge that we all know will never be discovered or invented in its totality. The wonderful part is that even knowing that, we continue creating and discovering... paradoxically motivated by our finiteness.